Random walk
Introduction
A random walk is a stochastic process which describes a path made of consecutive random steps.
Gaussian
In Gaussian random walk the steps follow a continuous Gaussian distribution. We will look at two different types, the univariate and multivariate kind.
Univariate
A univariate Gaussian Random Walk, is a series of i.i.d. \(\mathcal{N}(0,1)\) random variables such that
\[ \begin{align*} X_0&=0 \\ X_t&=X_{t−1}+\epsilon_t \end{align*} \]
Where \(t=1,2,\dots\) and \(\epsilon_t\) is a series of i.i.d. \(\mathcal{N}(0,1)\) random variables.
Let’s illustrate a simple univariate Gaussian random walk in Python, by plotting 1000 realisations.
import numpy as np
= 1000
N 23)
np.random.seed(= []
realisations for i in range(N):
=100))) realisations.append(np.cumsum(np.random.normal(size
import matplotlib.pyplot as plt
from plotutils import *
for i in range(N):
="k", alpha=0.1)
plt.plot(realisations[i], c"Univariate Gaussian random walk")
plt.title("t")
plt.xlabel("$x_t$")
plt.ylabel( plt.show()